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Ubc  mnivetsttp  ot  Cbtcago 


A  NEW  BASIS  FOR  THE 
METRIC  THEORY  OF  CONGRUENCES 

A  DISSERTATION 

submitted  to  the  faculty  of  the  ogden  graduate  school  of 

science   in   candidacy   for  the  degree  of  doctor  of 

philosophy 

Department  of  Mathematics 


BY 


LEVI   STEPHEN   SHIVELY 


Private  Edition,  Distributed  By 

THE  UNIVERSITY  OP  CHICAGO  LIBRARIES 

CHICAGO,   ILLINOIS 

1921 


EXCHANGE 


Ube  TDlntrersit^  ot  Cbtcago 

m  


A  NEW  BASIS  FOR  THE 
METRIC  THEORY  OF  CONGRUENCES 


A  DISSEKTATION 

submitted  to  the  faculty  of  the  ogden  graduate  school  of 

science   in   candidacy   for  the  degree  of  doctor  of 

philosophy 

Department  of  Mathematics 


BY 

LEVI   STEPHEN   SHIVELY 


Private  Edition,  Distributed  By 
THE  UNIVERSITY  OF  CHICAGO  LIBRARIES 

CHICAGO,   ILLINOIS 
1921 


St  \ 


A  NEW  BASIS  FOR  THE  METRIC  THEORY  OF 
CONGRUENCES. 

Introduction. 

In  the  classical  theory  of  rectilinear  congruences  as  developed  by 
Kummer/  the  analysis  is  based  upon  two  fundamental  quadratic 
forms,  formed  from  the  expressions  for  the  coordinates  of  the 
surface  of  reference  and  from  the  spherical  representation  of  the 
congruence.  Thus,  when  studied  by  this  method,  the  congruence 
is  in  reality  defined  by  its  relation  to  two  surfaces.  Important 
results  of  the  *Kummer  theory  are  the  existence  of  the  two  focal 
surfaces  and  of  two  one-parameter  families  of  developables.  These 
results  have  been  used  by  Wilczynski  as  the  basis  of  his  general 
projective  theory  of  congruences.^  In  the  present  paper  also,  the 
focal  surfaces  will  be  regarded  as  known  by  means  of  a  parametric 
representation,  such  that  the  ruled  surfaces  u  =  const,  and  v  = 
const,  of  the  congruence  are  its  developables. 

The  first  section  is  devoted  to  the  introduction  of  certain  local 
coordinate  systems,  all  closely  related  to  the  general  line  of  the 
congruence.  These  coordinate  systems  are  used  to  study  the  prop- 
erties of  the  congruence  in  the  neighborhood  of  one  of  its  lines,  as 
well  as  the  properties  of  its  Laplacian  transforms.  Homogeneous 
cartesian  coordinates  are  introduced  in  the  second  section  and  the 
equations  (DY  of  the  Wilczynski  theory  are  derived.  The  coef- 
ficients of  these  equations  being  expressed  in  terms  of  the  funda- 
mental quantities  of  the  focal  surfaces,  and  their  derivatives,  there 
is  here  afforded  a  convenient  means  for  investigating  the  relations 

^  Kummer,  AUgemeine  Theorie  der  geradlinigen  Strahlensysteme,  Crelle, 
Jour,  fiir  Mathematics,  57,  1860,  189-230. 

2  Wilczynski,  Sur  la  theorie  generale  des  congruences.  Memoires  publics  par 
la  Classe  des  Sciences  de  FAcademie  Royale  de  Belgique.  Bruxelles,  1911,  pp.  86. 
This  memoir  will,  in  what  follows,  be  referred  to  as  the  Brussels  Paper. 

3  Brussels  Paper,  p.  16. 

1 


46^853 


2         New  Basis  for  Metric  Theory  of  Congruences. 

between  the  focal  surfaces  and  the  congruence.  These  two  sections 
form  the  basis  for  the  present  investigation. 

In  the  third  section  we  obtain  new  proofs  of  some  of  the  most 
important  general  theorems  on  congruences.  These  results  are 
typical  of  those  obtained  by  means  of  the  Kummer  theory.  The 
fourth  section  takes  up  the  axis  and  ray  congruences  and  the  cor- 
responding curves  on  the  focal  surfaces,  all  of  which  were  recently 
discovered  by  Wilczynski.^  A  number  of  new  theorems  concerning 
these  configurations,  mainly  metric  in  character,  are  proven  in  this 
section.  In  the  final  section  we  discuss  the  equations  of  the 
Darboux  conies  of  a  point  on  one  of  the  focal  surfaces  and  consider 
some  special  cases. 

It  was  at  the  suggestion  of  Professor  Wilczynski  that  this  research 
was  undertaken  and  I  take  this  opportunity  of  expressing  my 
appreciation  of  the  helpful  encouragement  he  has  given  as  the 
work  progressed. 

1.  The  Congruence  Referred  to  Local  Coordinate  Systems. 

Let  the  focal  surfaces,  which  are  assumed  to  be  non-degenerate, 
be  denoted  by  Sy  and  Sz,  and  let  the  ruled  surfaces  u  =  const,  and 
V  =  const,  of  the  congruence  be  its  developables.  Then  Sy  will  be 
given  by  equations  of  the  form 

Vk  =  fk{u,  v)         {k  =  1,  2,  3), 

where  the  parametric  curves  u  =  const,  on  the  surface  Sy  are  the 
edges  of  regression  of  one  family  of  developables  of  the  congruence 
and  where  the  curves  v  =  const,  are  conjugate  to  u  =  const. 
Consequently,  making  use  of  the  customary  notation  of  the  theory 
of  surfaces,  we  have  D'  =  0  and  the  equations  of  Gauss^  for  Sy  will 
assume  the  form 

yuu  =  Aiyu-\-  A2yv  +  DX, 

(1)  Vuv  =  Biyu  +  Biyv, 

yvv  =  Ciyu  +  C2yv  +  D"X, 

1  Wilczynski,  The  General  Theory  of  Congruences,  Transactions  of  the  Ameri- 
can Mathematical  Society,  16,  1915,  311-327.  This  paper  will  hereafter  be  cited 
as  Gen.  Th.  Cong. 

2Bianchi,  Vorlesungen  iiber  Differential  geometrie,  p.  89.  Eisenhart,  A 
Treatise  on  the  Differential  Geometry  of  Curves  and  Surfaces,  p.  154. 


New  Basis  for  Metric  Theory  of  Congruences. 
where 


Ai  = 


11 
1 


i"l 


(2)  ft-ifl-ft-lz' 


o-^in  ^-i?!' 


are  the  Christoffel  symbols  formed  for  the  surface  Sy.  The  coor- 
dinates Zk,  {k  =  1,  2,  3),  of  P 2,  the  point  on  the  second  focal  surface 
Sz,  which  corresponds  to  P^  must  satisfy^  the  condition 


2  =2/ -^2/ 


'V) 


that  is 

(3)  yv  =  Bi(y  -  z), 

and  we  shall  have  Pi  ^  0,  since  otherwise  the  surface  Sy  would 
degenerate  into  a  curve.  Let  A[,  A'2,  -  -  - ,  C2  be  the  Christoffel 
symbols  for  the  surface  Sz,  and  let  us  denote  the  fundamental 
quantities  of  the  first  and  second  orders  for  Sy  and  Sghy  E''^\  •  •  •, 
Z)(^)''  and  E^'\  •  •  •,  D^'^''  respectively. 

The  first  Laplacian  transformed  congruence  consists  of  the 
tangents  to  the  curves  v  =  const,  on  Sy.  Let  its  second  focal 
surface  be  Sp.  Similarly,  let  S^.  be  the  second  focal  surface  of  the 
minus  first  Laplacian  transformed  congruence,  which  consists  of 
the  tangents  to  the  curves  u  =  const,  on  Sz.  By  the  same  method 
used  for  obtaining  (3)  we  find 

(4)  zu  =  B'^iz  -  y), 

(5)  '  yu  =  B2(y  -  p), 

(6)  z,  =  B[{z  -a). 
From  (3)  and  (1)  we  find  that 

/yN  _     {Bl)u    —    P1P2 

\' )  ^u  d2         yv) 

^  Eisenhart,  loc.  cit.,  p.  405,  eq.  43. 


4         New  Basis  for  Metric  Theory  of  Congruences. 

where  Bi  4=  0,  since  Sy  is  supposed  to  be  a  non-degenerate  surface. 
Eliminating  Zu  and  yv  from  (3),  (4)  and  (7)  gives 

(8)  B',  =  B,-^-^\ 

From  (5)  and  (I2)  we  find  in  the  same  way 

.o\  (^2)1;  —  B1B2 

(9)  Pv  = ^2 Vw 

By  means  of  (3),  (5)  and  (6)  we  find  the  following  distances 
between  corresponding  points  on  the  focal  surfaces: 


(10) 


We  have  also 


yP.= 

Bi 

yPp  = 

■ylE(y^ 
B, 

p  - 

Vg(^) 

B[ 


fiy)  jriz) 


COS  CO  =       ,     ,    .      ,  T  ,  cos  CO 


(11)  .  ^^^^  .      ,  H^'^ 

sm  CO  =     ,    ,  ,     — ,         sm  co 


■VE( !/)(;/( J/) '  ^jE(^z)Qiz)' 


^ff{y)   =   ^E{v)Q{y)  _  fiv)^^  JJiz)   =   ■^E^z)g(z)  —  7^(2)2^^ 

where  co  and  co'  are  the  angles  PzPyPp  and  PyPzP^  respectively, 
each  of  these  angles  being  so  defined  as  to  be  less  than  180°. 

We  shall  now  introduce  the  first  of  the  several  local  systems  of 
rectangular  coordinates  which  we  shall  use  subsequently.  This 
system,  which  will  be  designated  by  (I),  is  defined  as  follows.  The 
origin  is  the  point  Py.  The  {-axis  coincides  with  the  line  of  the 
congruence  through  Py,  so  directed  that  Pz  lies  on  its  positive  end. 
The  ?7-axis  is  in  the  tangent  plane  to  Sy  at  Py,  and  is  so  chosen 
that  the  angle  between  its  positive  end  and  PyPp  is  acute.  The 
^-axis  coincides  with  the  surface  normal  at  Py,  its  positive  direction 


New  Basis  for  Metric  Theory  of  Congruences.         5 

being  chosen  in  such  a  manner  that  the  relative  orientation  of  (I) 
is  the  same  as  that  of  the  original  axes. 

Referred  to  the  original  axes,  the  direction  cosines  of  the  axes 
of  (I)  are, 

iyk)v 


J-axis  : 


^iG^y)  ' 


(12)        ,;-axis  :  ^^J^^^  [.F^^Kyk).  -  G^^Kyk)u^ 


f-axis  :  —  Xk 


H(y) 


{yk+i)u    (yk+2)u 
(yk+i)v    (yk+2)v 

(k  =  1,2,  3;  2/4  =  yi,  y^  =  2/2). 

With  reference  to  (I)  the  lines  PyPp  and  PzPp  have  the  direction 
cosines 

F(y)  TJiv) 

P  P   •  0 

^    f"  '   '^E^y)Q(.v) '    ^Eiy)Q(y)  '      ' 

(13)  BiF^y^  -  B^G^y^    BiH^y^ 

^'^^  *  hi  '     hi    '  ^' 

-      (^1=  ^G^y\B\E^y^  -  2BiB2F^^  +  BlG^y^)), 
and  the  coordinates  of  P2  and  Pp  are 


(14) 


P.- (^^,0,0). 


p 


The  focal  planes  of  the  given  congruence,  passing  through  the  line 
PyPz,  are  the  tangent  plane  to  Sy  at  Py  and  the  osculating  plane 
of  the  curve  u  =  const,  at  the  point  Py.     Their  equations  are 

(15)  r  =  0,        and        D^y^' -slG^^^r)  -  CiH^y^^  =  0 
respectively;   the  direction  cosines  of  the  normal  to  the  latter  are 

,,,,     ^  D^yy'ylG^  CiH^y^ 

(16)  0, 

We  also  find  that  the  focal  planes  of  the  first  Laplacian  transform. 


6         New  Basis  for  Metric  Theory  of  Congruences. 

through  PyPp,  are  the  first  of  (15)  and  the  plane  through  Py  normal 
to  the  direction 

(17)  h       '  h~'  hi  ' 

The  axis  of  the  point  Pj,  is  the  line  of  intersection  of  the  planes 
which  osculate  the  curves  u  =  const,  and  v  =  const,  passing 
through  Py.  By  means  of  (16)  and  (17)  we  compute  the  direction 
cosines  of  the  axis  of  Py,  which  are  the  following: 

A3  '        h       '   '        h^ 

where 

;^3  =    JAlG^y^'D^yy'  -  2A2CiF^y^G^yW^yW^yy' 

+  ciE^y^G^yW^y^^-{-  G'^yW^y^^D^y^'^ , 

Hence  the  angle  between  PyPz  and  the  axis  of  Py  is 

_,  CiF^yW^y^  -  AiG^yW^yy' 

cos- ^^ ,^ 

while  the  angle  between  PyPp  and  the  axis  of  Py  is  found  to  be 


,  -ylG^y^CiE^yW^y^  -  A2F^yW^yy') 

cos  ^ p== . 

h-iE^y'> 

Since  a  congruence  is  normal  if  and  only  if  the  focal  planes  are 
perpendicular  we  have,  by  reference  to  (16)  and  (17)  the  following 
criteria : 

A  necessary  and  sufficient  condition  that  the  given  congruence 
be  normal  is  that  Ci  =  0;   and  a  necessarj^  and  sufficient  condition 
that  the  first  Laplacian  transformed  congruence  be  normal  is  that 
^2  =  0. 
And  similarly, 

A  necessary  and  sufficient  condition  that  the  minus  first  Laplacian 
transformed  congruence  be  normal  is  that  CI  =  0.^ 

1  These  criteria  obviously  agree  with  the  theorem  (see  Eisenhart,  loc.  cit., 
p.  401):  A  necessary  and  sufficient  condition  that  the  tangents  to  a  family  of 
curves  on  a  surface  form  a  normal  congruence  is  that  the  curves  be  geodesies. 


New  Basis  for  Metric  Theory  of  Congruences.         7 

The  axes  of  our  second  local  coordinate  system  (II)  are  the  lines 
of  curvature  tangents  and  the  surface  normal  at  Py.  In  order  to 
define  this  system  completely,  we  note  that  the  differential  equa- 
tions of  the  lines  of  curvature  on  ^S^  are 

dv  —  \idu  =  0, 

dv  —  \2du  =  0, 
where 

^  ~  r        /  TvG)^mv)myy'      i 

(19)  '  ^  ^  ^-^ 

X2  = 


14-^/l-J — — - 

^^\-L-r(^(j/)2)(i/)"_  G<<yW^y^yj' 

We  shall  refer  to  the  curves  through  Py,  defined  by  (18),  as  h  and  4 
respectively.  Taking  as  positive  that  direction  of  the  tangent  to  h 
whose  direction  cosines  are 

(y.)u  +  My.).  (it  =1,2,  3), 


and  denoting  by  6  the  angle  between  the  {-axis  and  the  positive 
tangent  to  h,  taken  in  the  direction  which  rotates  the  {-axis  through 
90°  into  the  ?7-axis,  we  have 

cos  d  = ,  —  =  a, 

^G^y)(E^y)  +  2\iF^y^  +  X'G^(^>) 
(20)  '      1        ^ 

smS  = =  B. 

■ylG^y)(E^y)  +  2Xii^(^>  +  X^G^y^) 

To  justify  the  choice  of  the  negative  sign  in  the  last  equation  we 
observe  that 

H(y) 
cos  (p  =  — 


^G^y\E^y^  +  2Xii'^(^>  +  \\G^y^) 


where  <p  is  the  angle  from  the  77-axis  to  the  positive  tangent  to  h 
measured  in  the  same  direction  as  6,  and  that  sin  B  =  cos  (p. 


8         New  Basis  for  Metric  Theory  of  Congruences. 

Let  the  positive  direction  of  the  tangent  to  I2  be  that  one  which 
agrees  with  the  positive  77-axis  when  system  (I)  is  rotated  about 
the  f-axis  through  the  angle  6.  Then  the  positive  fi,  771,  fi-axes 
of  system  (II)  are  defined  to  be  the  positive  tangents  to  h  and  hy 
and  the  positive  f-axis  respectively.  The  transformation  from  (I) 
to  (II)  is 

fi  =  a?  +  iSry, 

(21)  771  =  -  /5^  +  ary, 

Ti  =  r. 

Our  third  system  of  coordinates,  designated  by  (III),  is  obtained 
by  taking  as  origin  the  point  Pz  and  as  axes  the  lines  through  P  ^ 
which  are  parallel  to  the  corresponding  axes  of  (I).  If  the  coor- 
dinates of  the  point  (J,  77,  ^)  referred  to  (I),  be  (J',  -q' ,  ^')  when 
referred  to  (III),  we  have  the  following  equations  of  transformation: 


(22)  ,'  =  „  ' 

.     V  =  r. 

We  now  pass  to  the  system  (IV)  which  is  obtained  by  rotating 
(III)  about  the  J'-axis,  in  the  direction  from  the  ry'-axis  toward  the 
f'-axis,  through  that  angle  d\,  which  brings  the  ly'-axis  into  the 
plane  PyPzP^  in  such  a  manner  that  PzP„  projects  upon  the  positive 
end  of  the  ry'-axis.  The  new  positions  of  the  axes  of  (III)  are 
defined  to  be  the  J'',  f]",  r''-axes  of  (IV). 

.  The  plane  PyPzP^  is  both  the  tangent  plane  to  ^S  2  at  P  2  and  the 
osculating  plane  ofu=  const,  on  *Sy  at  the  point  Py.  The  direc- 
tion of  its  normal  is  given  by  (16).     Hence 


sin  ^1  =  € 


(23)  '  ' 

COS  01=  e    ,  ,  , 

■\Q{y)J)iv)"^  j^  C^H^y)^ 

where  e  has  one  of  the  two  values  d=  1.     The  equations  of  trans- 


New  Basis  for  Metric  Theory  of  Congruences. 
formation  from  (III)  to  (IV)  are 


(24)  "  

The  equations  which  enable  us  to  pass  from  (I)  to  (IV)  are 


(25)  V'  = 


e(r)CiH^y^  -  iD^yy'<G^) 


'\Q(.y)J[)(.yr'^  -|_  C2jj(y)2 
The  inverse  of  (25)  is 


?=  r  + 


(26) 


r 


B,    ' 

e(-  -n^D^yy^JG^^  +  ^CiH^y^) 


Since  the  line  P^P^.  makes  the  angle  180°  —  co'  with  the  positive 
J''-axis,  we  find  from  (10)  and  (11)  the  coordinates,  referred  to  (IV), 
of  P,  to  be  (-  P(^)/P;  -ylE^,  m^^B',  <W\  0).  Referred  to  (I), 
Pa  =  {fi,  ^2,  ^3)  where 


(27)  r2 


_  B[  VE(^)G^(^)  -  PiP(^) 
CiH^y^H^'^ 


B[^E^'\G^yW^yy'^ -{-  ClH^y^')' 


10       New  Basis  for  Metric  Theory  of  Congruences. 


J)(y)"JJ{z)  ^Q(y) 


n  = 


B[^E^'\G^yW^yy''' -{-  C\H^ 


2/)2 


We  shall  now  determine  the  ambiguity  in  the  sign  of  e.  Since 
the  lines  PzP„,  PzPy  and  the  surface  normal  of  Sz  at  Pz  whose 
direction  cosines  referred  to  the  original  axes  are 


(28)      - 


H(^) 


{^k+l)v        i^k+2)v 


(k  =  1,  2,  3;  2:4  =  zi,  Z5  =  Z2), 


have  the  same  relative  orientation  as  the  original  axes,  the  same  is 
true  of  the  J''-axis,  the  T^'^-axis,  and  this  direction  of  the  surface 
normal.  Consequently  the  sign  of  e  must  be  so  chosen  as  to  make 
the  f"-axis  coincide  with  this  direction  of  the  surface  normal. 

The  directed  line  whose  direction  cosines,  referred  to  (I),  are 
given  by  (16),  is  that  one  which,  referred  to  the  original  axes,  has 
the  direction  cosines 


(29) 


{yk+i)v 
iyk+i)v^ 


{yk+2)v 
{yk+2)vv 


(k  =  1,  2,  3;  2/4  =  yi,  2/5  =  ^2), 


where  h  is  a,  properly  chosen  positive  number.  Now  the  lines 
whose  directions  are  given  by  (28)  and  (29)  are  parallel.  Our 
problem  therefore  reduces  itself  essentially  to  that  of  determining 
whether  the  signs  of  corresponding  quantities  from  (28)  and  (29) 
are  the  same  or  opposite. 
From  (3)  and  (4)  we  obtain 


(30) 


Zu  ^  —  TT 


Bi^"' 


and  by  differentiating  this  and  (4)  with  respect  to  v,  eliminating  Zuv 
and  reducing  by  (3)  we  find 

1 


(31) 
where 


■^yv     n  yvv 


K=  1  + 


B2  \BiJv 


Upon  substituting  in  the  first  of  the  quantities  (28)  for  (2:2)  „,  (2^3) «, 


New  Basis  for  Metric  Theory  of  Congruences.       U 

(z2)v,  and  (z3)v,  their  values-  as  obtained  from  (30)  and  (31),  it 
reduces  to 

{y2)v     {yz)v 
{y2)vv    {yz)vv 

This  shows  that  the  directions  given  by  (28)  and  (29)  are  the  same 
or  opposite,  according  as  the  sign  of  B'2  is  negative  or  positive. 
For  a  non-degenerate  surface  Sz,  B'2  cannot  vanish.  Now  ac- 
cording as  e  =  +  1  or  —  1,  the  positive  f -axis  coincides  with  the 
directed  line,  whose  direction  is  given  by  (29),  or  the  opposite 
direction.  And  since  in  (IV)  the  t''-axis  coincides  with  the  surface 
normal  of  directions  (28)  it  follows  that,  according  as  B'2  is  negative 
or  positive,  e  takes  the  value  +  1  or  —  1. 

Our  last  system  of  local  coordinates  is  defined  by  its  relations  to 
the  lines  of  curvature  on  Sz.     The  equations  of  these  lines  are 

dv  —  \'idu  =  0, 
(32) 

dv  —  \2du  =  0, 

where  X'l  and  X2  are  formed  from  the  fundamental  quantities  for  Sz 
in  the  same  way  that  Xi  and  X2  are  formed  from  those  for  Sy  (see 
(19)).  In  what  follows  we  designate  those  curves  through  Pz 
defined  by  (32),  by  /!  and  /^  respectively.  Furthermore,  the 
positive  tangent  to  ![  is  defined  similarly  to  that  for  h.  Then, 
if  d^  denote  the  angle  from  the  positive  tangent  to  u  =  const,  on 
Sz  at  the  point  Pz,  to  the  positive  tangent  to  ![,  measured  in  the 
direction  from  u  =  const,  towards  v  =  const.,  we  have,  as  in  (20), 


cos  6'  = ,  ^- =  a 

ylG<^)(E^^)  +  2X;i^(^>  +  x;  (?<^)) 

sin  d'  = 


VG?(^)(^(^)-f-  2x;i^(^)  +  \?G^'^) 


Also,  if  (p'  be  the  acute  angle  from  the  ?7''-axis  to  the  tangent  to 
u  =  const.,  then 

cos  (p 


sm  (p 


12       New  Basis  for  Metric  Theory  of  Congruences. 

A  rotation  of  the  axes  of  (IV)  about  the  f ''-axis  through  the  angle 
ip'  -f  Q'  brings  the  f  and  ry^'-axes  into  coincidence  with  the  lines 
of  curvature  tangents,  the  positive  ?7"-axis  coinciding  with  the 
positive  tangent  to  l'\ .  The  respective  positions  of  the  axes  of  (IV) 
after  this  rotation  are  defined  to  be  the  axes  of  (V).     Let 

a"  =  cos  ((p'  +  (90  = p==--  , 

r  =  sin  {<p'  +  6')  =  ,  ^-. 

Then  if  we  denote  by  {^i,  rji,  fl')  the  coordinates,  referred  to 
(V),  of  the  point  whose  coordinates  referred  to  (IV)  are  (^'\  rj'',  ^'') 
we  have 

fr  =  a''r  +  r>?", 

(33)  vi  =  -  P"^"  +  a'W\ 

2.  The  Differential  Equations  (D)  of  the  Congruence. 

Wilczynski  has  shown^  that  the  projective  theory  of  congruences 
may  be  based  upon  a  completely  integrable  system  of  partial 
differential  equations  of  the  form 

yv  =  mz,        Zu  =  ny, 
(D)  yuu  =ay-\-hz-\-  cyu  +  dzy, 

Zvv  =  a'y  +  h'z  +  c'yu  +  d'zv. 

He  has  shown  that  such  a  system  has  four  pairs  of  linearly  inde- 
pendent solutions  {yk,  Zk),  (k  =  1,  2,  3,  4),  and  that  if  (yi,  •  •  •,  2/4) 
and  {zi,  •  •  • ,  24)  be  regarded  as  the  homogeneous  coordinates  of 
two  points  Py  and  Pg,  the  locus  of  these  points,  as  u  and  v  vary, 
will  be  the  focal  surfaces  Sy  and  ^Sz  of  the  congruence.  It  is  the 
object  of  this  section  to  obtain  the  differential  equations  (D)  of  our 
congruence,  with  coefficients  expressed  in  terms  of  the  fundamental 
quantities  of  the  focal  surfaces  and  their  derivatives.  We  shall 
^  Brussels  Paper,  pp.  9-19. 


New  Basis  for  Metric  Theory  of  Congruences.       13 

exclude  from  present  consideration  those  congruences  whose  lines 
are  the  tangents  to  asymptotic  curves  on  either  of  the  focal  surfaces. 
In  our  notation,  this  is  expressed  by  the  conditions,  D^^)"  y>^  0 
and  Z)(^>  9^  0. 

We  can  then  eliminate  X  and  y^v  from  the  first  and  third  of 
equations  (1)  and  the  equation  obtained  by  differentiating  (3)  with 
respect  to  t.     This  gives 

(34)  yuu=  ^^{y  —  z)^  ciyu  +  diz^y 
where 

7)(y) 

ai  =  A,B,  +  ^(^.  [(5i  -  C^)B,  +  (5i)  J, 

Proceeding  in  a  similar  manner  with  the  corresponding  equations 
for  the  surface  »Sz  we  obtain 

(35)  Zvv  =  cl[{z  —  y)  -\-  c[yu  +  d[zv, 
where 

a[  =  C[B',  +  -^^  [(5;  -  A[)B',  +  (5;)  J, 

^1    -     ~     J)iz)    ^2, 


di  —  62        T\( y\  A2. 


Let  now  {yi,  2/2,  y^,  1)  and  (21,  Z2,  zz,  1)  be  homogeneous  cartesian 
coordinates  of  the  points  Py  and  Pz  respectively.  The  system  of 
equations  consisting  of  (3),  (4),  (34)  and  (35)  is  satisfied  by  these 
coordinates  since  it  is  obviously  satisfied  by  2/  =  1,  s  =  1.  To 
obtain  from  these  equations  the  system  (D)  we  set 

V  =  fBidv, 
q  =  fB'zdu, 


14       New  Basis  foe  Metric  Theory  of  Congruences. 

and  make  the  transformation 

y  =  e^y, 
z  =  e^z. 
Then  (3),  (4),  (34)  and  (35)  become 

yv  =  mz,         Zu  =  ny, 

(36)  yuu=  ay-\-bz-\-  cyu  +  dz^y 

Zvv  =  a'y  +  h'z  +  c'yu  +  d'z^, 
where 

n=  -  e^^B2, 

a  =  A2B1  +  ^^[(5i  -  C2)B,  +  (Bi),  -  CpJ 

+   AiPu  —   pi   —   Puuy 

(37)  ^=-'''d^'^^' 

a'  =  -  e^^  ^B',C[  +  ^  { (5;  -  A[)B', 

+  {B',)u+B',pu]'^, 

^'  =  ^^c;  +  ^[(5;  -  A\)B'2  +  (5;).  -  ^;gj 

d   =  C2        jyjy  A2       2qv. 

Equations  (36)  constitute  the  system  (D).     They  are  satisfied  by 
corresponding  pairs  of  the  homogeneous  coordinates  {e~^y\,  e~^y2, 


New  Basis  for  Metric  Theory  of  Congruences.       15 

e~^yz,  e~P)  and  (e~^Zi,  e~%,  e~^Z3,  e~^)  of  the  points  Py  and  Pz 
respectively. 

3.  Some  of  the  Principal  Results  of  the  Classical 
Theory  of  Congruences. 

This  section  will  be  devoted  to  the  proof  of  a  few  of  the  general 
theorems  on  congruences  which  are  ordinarily  proved  by  means 
of  the  Kummer  theory.  Use  will  here  be  made  of  the  coordinate 
systems  which  were  established  in  section  1. 

Let  Py  and  Pz  be  points  in  the  neighborhoods  of  Py  and  Pg 
obtained  by  giving  to  u  and  v  the  increments  8m  and  8v  respectively. 
Then  by  Taylor's  theorem,  (3),  (4),  (5),  and  (6), 

Y  =  y-\-  yubu  +  y^bv  +  •  •  • 
=  y-\-  B2(y  -  p)bu  +  Bi{y  -  z)8v  +  •  •  • 

Z  =  z  +  Zu8u  +  Zv8v  +  •  •  • 

=  z  +  B^iz  -y)  +  B[{z  -  a)8v  +  •  •  -, 

where  the  terms  not  written  are  of  higher  order  than  the  first  in 
du  and  8v.  Denoting  by  ({,  tj,  0  and  (^i,  rji,  fi)  the  coordinates  of 
Py  and  Pz  when  referred  to  (I),  these  equations,  together  with  (14) 
and  (27)  give,  up  to  terms  of  the  second  order, 

J  =  a8u  +  P8v,         Ji  =  ao  +  oii8u  +  ^i8v, 

7j  =  y8u,  rji  =  yi8v, 

r  =  0,  ri  =  8i8v, 

where 


V^'  '  B, 


^G'^y^'  V^(z)((^(?/)2)(2/)"^-|-  C^H^y^^) 


«o  =  —B — ,  8i=  e 


^1       '  V^(^)((;f(2/)2)(y)"^+    (72^(^)2)  * 


16       New  Basis  for  Metric  Theory  of  Congruences. 

The  line  which  is  the  common  perpendicular  to  PyPz  and  PyPi 
has  direction  cosines  proportional  to 


(38)  0,  8it,  y  -  yit; 


(-S) 


and  the  abscissa  of  the  foot  of  this  common  perpendicular  upon  the 
latter  line  is 

«o7(7  —  TiO 


(39)  ?  = 


{y-7ity+dlt' 


In  order  to  find  the  values  of  t  for  which  this  abscissa  is  a  maximum 
or  a  minimum,  we  differentiate  with  respect  to  t  and  set  the  result 
equal  to  zero.     This  gives 

2 

(40)  t'-2^i  +  Z^^=0' 

7i         7i  +  Oi 

which  is  the  differential  equation  of  the  curves  in  which  the  principal 
surfaces  of  the  congruence  intersect  the  focal  surface  Sy.     Solving 

(40)  we  obtain 

(41)  ^  J 

^^  =  ^(i- vwtl)' 


If  we  use  these  values  of  t  in  (39)  we  obtain  the  abscissas  of  the 
limit  points  of  the  line  PyPz.     They  are 

Q!o7i^ 

(42) 

The  midpoint  of  the  segment  joining  these  limit  points  has  the 
abscissa  |q;o  =  ^('sG^/Bi).  This  proves  the  well-known  the- 
orem :^ 

The  midpoints  of  the  two  segments  bounded  respectively  by  the  limit 
points  and  by  the  focal  points  coincide. 

^  This  theorem  and  the  others  of  this  section,  together  with  Hamilton's  equa- 
tion, are  given  in  sections  4  and  5  of  the  previously  cited  memoir  by  Kummer. 


New  Basis  for  Metric  Theory  of  Congruences.       17 

Again,  if  the  values  of  h  and  ^2  from  (41)  be  substituted  suc- 
cessively in  (38)  the  directions  of  the  common  perpendiculars  at 
the  limit  points  are  obtained.  It  is  easy  to  verify  that  these  direc- 
tions are  perpendicular.  Since  these  perpendiculars  and  the  line 
of  the  congruence  to  which  they  are  normal  determine  the  principal 
planes,  we  have  the  theorem : 

The  two  principal  planes  through  each  line  of  the  congruence  are 
perpendicular. 

Let  us  denote  by  df  and  di  the  distances  between  the  focal  points 
and  the  limit  points  respectively.  Then  df  =  VCr^^V^i-  Also 
from  (42) 


^G(y)D^yy''  +  C'.H^yy' 


Therefore 


Q(y)J){yy 

df  j)^yy''<G^y'> 


di       ^Q{v)D{v)"^  _|_  clH^y^^ 


sm  r 


by  (15)  and  (16),  where  r  is  the  angle  between  the  focal  planes. 
This  gives  the  theorem: 

The  ratio  of  the  distance  between  the  focal  points  to  the  distance 
between  the  limit  points  is  equal  to  the  sine  of  the  angle  between  the 
focal  planes. 

In  particular,  when  the  congruence  is  normal,  Ci  =  0  and  this 
ratio  is  unity.  Thus,  the  limit  points  coincide  with  the  focal 
points  in  a  normal  congruence  and  conversely. 

In  order  to  deduce  Hamilton's  equation,  let  co  be  the  angle  between 
the  common  perpendiculars  whose  directions  are  given  by  (38)  for 
t  =  t  and  t  =  ti.     The  calculation  shows  that 


f2 

=   COS^ 

CO, 

which  is  equivalent  to 

?  = 

■■    f  1  COS^  CO  +    ^2 

sin^ 

CO, 

This  is  Hamilton's  equation. 

18       New  Basis  for  Metric  Theory  of  Congruences. 

4.  The  Axis  and  the  Ray  Congruences. 

The  developable  surfaces  of  the  axis  congruence  intersect  ^^  in 
the  axis  curves.  Since  Sy  is  not  degenerate,  m  9^  0,  and  thus 
these  curves  are  determined  by  the  differential  equation '} 

(43)  -5^2  _  j5^5^  _  c'ddv^  =  0. 

m 

In  terms  of  the  fundamental  quantities  of  the  focal  surfaces,  the 
coefficients  of  (43)  are  found  by  (37)  and  the  definitions  of  c^i  and  / 
to  be 


'~]^B,B,  +  2{B,)u-  (^1), 


+ 


l^^C^'^lJ^-    (^2).-^^l0g;^(^,J, 

(44)       I  =  A^Bi  +  ^,  (Bl  -  B,C,  +  (5i),  -  B^C^) 

+  A,B,  -  {B2)uy 
c'd  =  B,B,  +  {B,)u  -  (^1).  +  [^(yy'C^)^' 

A  necessary  and  sufficient  condition  that  the  axis  curves  be  conju- 
gate is 


which  reduces  to 


m 


Q2    ,      D(i/) 


(45)  (5i)"-(52).-^-^-^log5^,=  0. 

If  we  confine  our  attention  to  those  congruences  whose  focal  surfaces 
are  distinct  so  that  Z>^^^  as  well  as  D^^^'  is  different  from  zero, 
the  Codazzi  equations^ 

Z)?^  -  5i7)(^>  +  A^B^yy'  =0, 
(46) 

Z)ir>"  +  CiD^y^  -  B^B^yy  =  0, 

1  Gen.  Th.  Cong.,  p.  316.     See  eqs.  (7),  p.  315  for  the  definitions  of  di  and  I. 

2  Eisenhart,  loc.  cit.,  page  155. 


New  Basis  for  Metric  Theory  of  Congruences.       19 

enable  us  to  express  (45)  by  the  equation 

(  ,  ^^\    _(  n  P^\ 
K^^D^y^  )u~\   'B^yy).' 

This  condition  is  obviously  satisfied  when  the  congruence  and  its 
first  Laplacian  transform  are  both  normal.  But  this  is  equivalent 
to  the  condition  that  ^Sj,  be  a  surface  of  Voss.  Hence  we  have  the 
theorem : 

The  axis  curves  with  respect  to  a  congruence  consisting  of  the  tangents 
to  one  of  the  two  families  of  conjugate  geodesies  on  a  non-developable 
surface  of  Voss,  are  conjugate. 

We  shall  also  prove  that 

If  a  congruence  consists  of  the  tangents  to  the  lines  of  curvature  of 
one  system  on  a  surface  Sy,  this  surface  being  either  a  quadric,  or  a 
surface  of  revolution  of  constant  total  curvature,  then  the  axis  curves 
are  conjugate.  To  prove  this  theorem  we  observe  that,  in  case 
F<iy)  =  0,  (45)  becomes 

Jliy)  Jmy) 

— — — — =  U'V 

B^yy^G^y^  ' 

where  C7  is  a  function  of  u  alone  and  F  is  a  function  of  v  alone. 
This  condition  is  clearly  satisfied  by  an  isothermic  surface  whose 
lines  of  curvature  are  isothermal-conjugate.  The  fundamental 
quantities  of  the  quadrics^  show  that  they  have  these  properties. 
Furthermore  since  all  surfaces  of  revolution  are  isothermic,  and 
since  the  lines  of  curvature  of  all  surfaces  of  constant  total  curvature 
are  isothermal-conjugate,  the  part  of  the  theorem  relating  to 
surfaces  of  revolution  of  constant  total  curvature  also  follows. 
The  foci  of  the  axis  of  Pj,  are  given  by  the  factors  of^ 

c'dxi  -  mlyr^y^  -  dmT^y^\ 

the  point  P^iv)  being  the  intersection  of  the  axis  with  the  line  PzP„. 
These  foci  are  on  the  same  or  on  opposite  sides  of  Pj,  according  as 
—  (dm/c'di)  is  positive  or  negative.     The  fifth  of  the  integrability 

1  Ibid.,  pp.  239-241. 

2  Gen.  Th.  Cong.,  eq.  (10),  p.  316. 


20       New  Basis  for  Metric  Theory  of  Congruences. 
conditions^  of  system  (D)  gives  the  equation 

IB1B2  -  (5i)  J  (^  1  -  p(y)vp(,)  j 

If  we  consider  the  case  in  which  the  axis  curves  are  conjugate  and 
use  (48)  the  value  of  —  (dm/c'di)  reduces  to 

2)(y) 


^B^B,+  {BOu-{A,).-\-(^^.C,^J 

This  is  positive  or  negative  according  as  D^^^  and  D^^)"  have 
opposite  or  the  same  signs.     Hence  we  have  proved  the  theorem: 

//  the  axis  curves  on  a  surface  Sy  are  conjugate,  the  foci  of  the  axis 
lie  on  the  same  or  on  opposite  sides  of  Py  according  as  the  total  curva- 
ture of  Sy  at  Py  is  negative  or  positive. 

We  next  proceed  to  a  consideration  of  the  ray  curves  on  Sy. 
Their  differential  equation  is^ 

(49)  dndu'  -  Uubv  -'!^di^=0. 

m 

In  this  equation  I  is  given  by  (44),  and 

^^  =  5ur'[^i^2  -  (5i)«], 
(50) 

—  =  B1B2  —  {B^v- 

ifv 

Thus  we  see  that  if  D^^W^^^"  ?^  0,  a  necessary  and  sufficient  con- 
dition that  the  ray  curves  be  conjugate  is 

(51)  {B,)u  ==  (52).. 

Now  51^2  —  iBi)u  and  B1B2  —  {B2)v  are  the  Laplace-Darboux 

1  Brussels  Paper,  p.  17.    See  also  Gen.  Th.  Cong.,  p.  313. 

2  Gen.  Th.  Cong.,  p.  318. 


New  Basis  for  Metric  Theory  of  Congruences.       21 

invariants^  of  the  equation  (I2).  This  gives  the  result  found  by 
Wilczynski  •? 

"  A  conjugate  system  on  a  non-developable  surface  has  equal 
Laplace-Darboux  invariants  if  and  only  if  its  ray  curves  also  form  a 
conjugate  system." 

Let  us  interpret  the  condition  for  conjugate  ray  curves  in  the 
case  in  which  the  original  parametric  net  on  8y  consists  of  the 
lines  of  curvature.     Then  (51)  becomes 

where  TJ  and  V  are  functions  of  u  alone  and  v  alone  respectively. 
But  this  condition  is  satisfied  when  and  only  when  the  surface  Sy 
is  isothermic.     Hence 

A  necessary  and  sufficient  condition  that  the  ray  curves  with  respect 
to  the  lines  of  curvature  on  a  non-developahle  surface  be  conjugate,  is 
that  the  surface  be  isothermic. 

Wilczynski  has  shown  that  a  necessary  and  sufficient  condition 
that  the  axis  curves,  for  a  non-degenerate  surface  Sy,  coincide  with 
the  original  conjugate  net  on  ^S^,  is  that  Sz  be  developable.  He 
shows  further  that  when  this  condition  is  satisfied  it  also  follows 
that  Sp  is  developable.  We  shall  consider  the  corresponding 
question  for  the  ray  curves.  Reference  to  (50)  shows  that  when 
j)iv)D{yy'  ^  Q  the  ray  curves  are  parametric  when  and  only  when 
the  invariants  B1B2  —  {Bi)u  and  B1B2  —  {B2)v  are  both  zero. 
Since  these  are  the  conditions  that  Sz  and  Sp  degenerate  into  curves, 
we  obtain  the  following  result : 

Necessary  and  sufficient  conditions  that  the  ray  curves  on  a  non- 
developable  surface  Sy  coincide  with  the  parametric  curves  are  that  the 
surfaces  Sz  and  Sp  be  degenerate. 

To  determine  whether  the  foci  of  the  ray  of  Pj,  lie  on  the  same 
or  on  opposite  sides  of  P  3  we  proceed  similarly  to  the  method  used 
for  the  corresponding  problem  of  the  foci  on  the  axis.  Here  we 
find  that  they  are  on  the  same  side  or  on  opposite  sides  of  Pa  ac- 

J  Darboux,  Lemons  sur  la  Theorie  Generale  des  Surfaces,  Tome  2,  pp.  23-25. 
2  Gen.  Th.  Cong.,  p.  319. 


22       New  Basis  for  Metric  Theory  of  Congruences. 

cording  as  —  (mn/dnii)  is  positive  or  negative.^     When   (51)   is 
satisfied 

2)(y)" 

^2(p-3)  ^ 


drrii  B\ 

Therefore 

If  the  ray  curves  on  Syhe  conjugate,  the  foci  of  the  ray  of  Py  lie  on 
the  same  side  or  on  opposite  sides  of  Pz  according  as  the  total  curvature 
of  Sy  at  Pyis  negative  or  positive. 

Reference  to  (45)  and  (51)  shows  the  truth  of  the  following  the- 
orem: 

Any  two  of  the  three  conditions:  (a)  the  axis  curves  on  Sy  are 
conjugate,  (b)  the  ray  curves  on  Sy  are  conjugate,  (c)  the  parametric 
curves  onSy  are  isothermal-conjugate;  for  a  non-developable  surface  Sy, 
imply  the  third. 

For  further  investigation  of  the  case  in  which  the  conjugate  net, 
in  which  Sy  is  met  by  the  developables  of  the  congruence,  consists 
of  the  lines  of  curvature,  we  shall  have  need  for  the  Gauss  and 
Codazzi  equations.     In  this  case  they  are^^ 

:^)„-^(VG-)„  =  o. 

Let  us  consider  a  congruence  which  has  the  property  that  it  and 
its  first  Laplacian  transform  are  both  normal.  If,  in  addition,  the 
lines  of  curvature  are  parametric,  we  then  have  E^p  =  6r?^  =  0. 
With  these  conditions  the  first  of  (52)  gives  D^^W^y^"  =  0;  hence 
Sy  is  developable.     We  may  state  this  result  as  follows: 

//  a  congruence  and  its  first  Laplacian  transform  are  both  normal 

1  See  eq.  (22),  Gen.  Th.  Cong. 

2  Bianchi,  loc.  cit.,  p.  94.     Eisenhart,  loc.  cit.,  p.  157. 


New  Basis  for  Metric  Theory  of  Congruences.       23 

and  they  meet  their  common  focal  surface  in  its  lines  of  curvature  then 
that  surface  is  developable} 

The  converse  of  this  theorem  is  not  true.^  However  the  truth  of 
the  following  theorem,  which  resembles  somewhat  the  converse  of 
the  preceding,  will  be  established. 

If  a  surface  is  developable,  then  one  of  the  two  congruences  which 
consist  of  the  tangents  to  the  two  systems  of  curves  of  a  conjugate  net 
on  the  surface,  is  a  normal  congruence. 

Taking  the  given  conjugate  curves  as  parametric  we  have  one  of 
the  two  cases:  Z)(^>  =  0,  D^y^  9^  0;  D^y^  9^  0,  D^^^"  =  0.  In  the 
first  case  (46i)  gives  A2  =  0,  showing  that  tangents  to  the  curves 
V  =  const,  form  a  normal  congruence.  And  in  the  second  case 
(462)  gives  Ci  =  0  with  the  result  that  the  congruence  consisting 
of  the  tangents  to  the  curves  u  =  const,  is  normal. 

5.  The  Darboux  Conics. 

The  surfaces  upon  which  the  conjugate  system  determined  by  the 
developables  of  the  congruence  have  equal  Laplace-Darboux  invari- 
ants have  been  studied  by  both  Darboux  and  Wilczynski.  We 
have  already  seen  the  interesting  geometrical  interpretation  of  this 
property  by  the  latter,  namely,  the  conjugacy  of  the  ray  curves. 
Although  Darboux's  interpretation  is  entirely  different  from  that  of 
.Wilczynski,  the  two  are  essentially  equivalent.  Darboux's  theorem 
is  that  if  these  invariants  are  equal,  there  exists  in  the  tangent  plane 
to  Sy  Sit  Py  Si  conic  having  second  order  contact  at  Pz  with  the 
curve  V  =  const,  and  having  also  second  order  contact  at  Pp  with 
the  curve  u  =  const.,  and  conversely.  Whether  these  invariants 
are  equal  or  not,  there  are  two  conics  (distinct  if  the  invariants  are 
unequal)  which  Wilczynski  has  called  the  Darboux  conics  of  Py, 
the  first  of  which  has  second  order  contact  with  v  =  const,  at  Pz 

^  The  excluding  from  consideration  of  surfaces  for  which  D^^^"  =  0  is  not 
vital  to  this  argument. 

2  That  there  are  surfaces  for  which  the  hypothesis  of  the  converse  is  satis- 
fied is  shown  by  the  example : 

E(y)  =  a,      F^y^  =  0,       G^y^  =  v?f{v),      D^y^  =  0,       D^y^  =  0,       D^y^'  =  u, 

a  being  a  constant,  not  zero,  and  f{v)  an  arbitrary  function  of  v  alon  e,  not  iden- 
tically equal  to  zero. 


24       New  Basis  foe  Metric  Theory  of  Congruences. 

and  first  order  contact  with  u  =  const,  at  Pp;  and  the  second  has 
second  order  contact  at  P^  with  u  =  const,  and  simple  contact  at 
Pz  with  V  =  const.  We  shall  hereafter  refer  to  these  as  the  first 
and  second  Darboux  conies  of  the  point  Py  with  respect  to  the  given 
conjugate  net  on  the  surface  Sy, 

Let  us  proceed  to  find  the  equation  of  the  first  Darboux  conic 
of  Py.  Denoting  by  Pz  the  point  near  Pz,  obtained  by  giving  to  u 
the  increment  8u,  we  have  by  (4)  and  (5) 


(53) 


=  z+B',(z-y)8u-h  hL{iB2)u+B2]{z-y) 

Let  (J,  77,  f)  be  the  coordinates,  referred  to  (I),  of  Pz-    Then  from 

(53)  and  (14), 

(54)  ^=  vM-\-  ••., 


where 


1  ^G^y^  f  _,,     .    ^,2  .  B^B'^F 


T  =  o 


2    Bi 
1  BoH^y^ 


[mu+B'U^-^\, 


V  = 


2    V^(y)  * 
If  we  impose  upon  the  conic 

(55)  aif  +  2A1J77  +  61772  _^  2g,^  +  2/177  +  Ci  =  0 

the  condition  that  it  be  satisfied,  up  to  and  including  terms  of  the 
second  order  in  8u,  by  (53)  we  find,  upon  expressing  ^1,  /i  and  Ci 
in  terms  of  ai,  hi  and  61  that  it  reduces  to 

(56)  aif  +  2^1^77  +  61772  -  2aiQ:{  "  (  ^'  +  2^iq:  ^  77  +  aio"  =  0. 


New  Basis  for  Metric  Theory  of  Congruences.       25 

We  shall  now  determine  the  ratios  ai  :  h  :  61  such  that  (56)  has 
first  order  contact  with  the  cuspidal  edge  at  Pp.  Denote  by  Pr 
the  point  near  Pp  obtained  by  giving  to  v  the  increment  8v.  Then 
by  (3),  (5)  and  (9), 

R=  p-[-  Pv8v  +  ~^dv^+     ", 

^^^)         =  p  +  ilf (p  -  y)bv  +  ^ { (if.  +  M^){p  -  y) 

where 


M  =  Bi- 


B, 


Let  (Ji,  ryi,  ^1)  be  the  coordinates,  referred  to  (I),  of  Pr.     Then  it 
follows  from  (57)  and  (14)  that 

fi  =  «i  +  iSi5i)  +  yi^v^  +  •  •  •, 
(58)  'r]i  =  \i-{- ^ihv-\-  vih^-\-  ••  •, 

fi=  0+..., 


where 


JP(y) 

""'^b^Tg^' 

1 


7i  =  2l 


Xi  = 


(M,  +  M2)  ■;^^-y=^+  M^lG^y^  I , 


b^4g^^ 


Ml 


b^4g^^' 

_MPM_ 

b.4W^' 


H(y) 

Pi  =  UM.  +  M') 


B2<G^y^ 

Now  let  ai  -.hi  :  61  be  such  that  (56)  is  satisfied  up  to  and  including 
first  order  terms  in  bv,  by  (58).     Thus  the  equation  of  the  first 


26       New  Basis  for  Metric  Theory  of  Congruences. 


Darboux  conic  of  Py  is  (55)  where  the  coefficients  are  given  by 

h  =  my^[G^y\Bi)u  -  BlF^y^'], 

b,  =  BlF^y^^  -  2{Bi)uF^yW^y^  +  BlG^y^\ 

(59)     •  , 

gi=  -  BiH^y^'^lG^, 

/i  =  myWG^\BiF^y^  -  B^G^y^), 
ci  =  G^y^H^y^\ 

In  a  similar  manner  we  find  the  equation  of  the  second  Darboux 
conic  of  Py.  It  is  found  to  differ  from  that  of  the  first  only  in  the 
values  of  hi  and  6i.  The  values  of  these  coefficients  for  the  second 
conic  are  obtained  from  (59)  by  changing  {Bi)u  to  {B2)v  This 
agrees  with  the  fact  that  the  two  conies  coincide  when  and  only 
when  the  Laplace-Darboux  invariants  are  equal.  To  the  results 
which  are  deduced  in  the  following  paragraphs  for  the  first  Darboux 
conic,  there  evidently  correspond  similar  results  for  the  second, 
which  can  be  stated  analytically  by  changing  (J5i)„  to  iB2)v  in  the 
expressions  for  those  of  the  first  conic. 
From  (59)  we  find 

aj)i  -  hi  =  G^y^'H^y^'lBlBl  -  (5i)3. 

Hence  the  species  of  the  conic  depends  upon  the  invariant  B1B2 
—  {Bi)u  and  upon  B1B2  +  (5i)u.  If  the  invariant  B1B2  —  {Bi)u 
is  not  zero,  that  is  if  iSz  is  not  degenerate,  the  conic  is  a  parabola 
when  and  only  when  B1B2  +  {Bi)u  =  0.  The  conditions  that  it 
be  a  circle  reduce  to 

El      Gl 


^  =  7rV0, 


(60) 


The  determinant 


E 

{Bi)u 
Bl 


G 

F 
G' 


A  = 


h 
bl 

h 


-  G^y^'H^y^'lBiB2  -  (Bi)uJ 


vanishes  when  and  only  when  B1B2  —  {Bi] 
theorem, 


vanishes.     Hence  the 


New  Basis  for  Metric  Theory  of  Congruences.        27 

A  necessary  and  sufficient  condition  that  the  first  Darboux  conic  of 
the  point  Py  degenerate  into  a  pair  of  straight  lines  is  that  the  focal 
surface  be  a  curw. 

When  the  condition  of  this  theorem  is  satisfied  the  equation  of  the 
conic  becomes 

[5iF(^>  J  -  {BiF^y^  -  B2G^y^)v  -  H^'^  V^J  =  0. 

This  is  the  equation  of  the  line  PzPp,  the  ray  of  Py. 

In  case  F^^^  =  0,  the  equation  of  the  first  Darboux  conic  reduces 
to 

(61)     BlE^y^e  +  2  <¥^^G^\Bi)uiri  +  BlG^y^-q^ 

-  2BiE^y^  ^W^i  -  2B2  ^E^G^y^T]  +  E^y^G^y^  =  0. 

.       /      B2^lG^^  Bi^Wy^      \ 

Its   center  is   the  pomt   [b^b2+(B,).^      B^B.+  iBOuJ'       ^^ 

{Bi)u  =  0,  that  is,  since  i^^^^  =  0,  if  E^^^  is  the  product  of  a  function 
of  u  alone  by  a  function  of  v  alone,  the  axes  of  the  conic  are  paralled 
to  the  lines  of  the  given  congruence  and  of  its  first  Laplacian  trans- 
form which  intersect  at  Py.  Then  aA  -  h\  =  E^^^G^^^BlBl 
and  the  conic  is  an  ellipse. 

By  means  of  the  Gauss  and  Codazzi  equations  it  may  be  shown 
that  there  is  no  surface,  aside  from  the  trivial  case  of  the  plane, 
for  which  the  first  and  second  Darboux  conies  of  every  point  with 
respect  to  the  lines  of  curvature,  are  coincident  circles.  For, 
suppose  they  are  both  circles  and  that  they  coincide.  The  first 
of  (60)  implies  the  existence  of  a  function  (p{u,  v)  such  that 

£'">  =  L>Pu  +  fiu)J,        G  =  [±  ^.  +  g{v)J 

where  f{u)  and  g{v)  are  arbitrary  functions  of  u  and  v  respectively. 
Now,  by  the  second  of  (60)  and  the  coincidence  of  the  conies, 


[(^^1    _[{G^-\    _. 


But  these  last  conditions,  together  with  (52i)  show  that  D^^W^y^" 
=  0.  Suppose  D^y^  =  0.  Then  by  (522),  since  (V^),  9^  0,  it 
follows  that  D^y^"  =  0.  Similarly  it  may  be  shown  by  (523)  that 
D^yy  =  0  implies  D^y^  =  0. 


VITA. 

Levi  S.  Shively  was  born  near  Cerro  Gordo,  Illinois  on  October 
10,  1884.  His  early  education  was  received  in  the  public  schools 
of  Cerro  Gordo.  After  spending  several  years  in  Mt.  Morris 
College  he  entered  the  University  of  Michigan  in  1906,  from  which 
he  graduated  two  years  later  with  the  degree  Bachelor  of  Arts. 
During  the  years  1908  to  1915  he  was  Professor  of  Mathematics 
in  Mt.  Morris  College.  He  entered  The  University  of  Chicago 
for  graduate  study  in  the  summer  of  1915  and  during  nine  succeeding 
quarters  was  a  student  of  Professors  Moore,  Dickson,  Bliss,  Wilczyn- 
ski,  Lunn,  Moulton,  MacMillan  and  Birkhoff.  He  received  the 
degree  Master  of  Arts  in  the  summer  of  1916,  the  thesis  for  same 
having  been  written  under  the  direction  of  Professor  Moore.  The 
present  thesis,  in  candidacy  for  the  degree  Doctor  of  Philosophy 
was  prepared  under  the  direction  and  inspiration  of  Professor 
Wilczynski. 


28 


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